OK. They appear to be the traditional type with a rubber washer. They are easily repaired. New washers cost pence. you probably need a standard 3/4" for the bath tap. Most basin taps are 1/2". There are a few rare odd sizes.
Quite likely the washer has worn enough that the internal brass jumper has reached the end of its screw thread and is mispositioned. If you look up "changing a tap washer" you willl find billions of youtube clips.
Tap washers usually wear out if they are excessively tightened with great force. This is usually done by an older person who grew up with worn out taps. My own taps turn with a finger tip and I have never worn one out.
Things I recommend:
clean the parts with a green nylon pan scourer to remove limescale and tarnish
try to get some white silicon plumbers grease to lubricate the moving parts before you reassemble them
buy some plumbers PTFE tape to put on the brass threads of the cartridge before you screw it into the base of the tap. it seals against leaks and prevents the thread seizing in future.
Put in the plug, and lay a large old towel in the bath to catch the tools and parts you drop, and prevent them scratchng the bath and falling down the plug hole.
if you can get the hang of re-washering a tap you will find it so easy and pleasurable that you will do all the taps in the house, and try to talk friends and neighbours into letting you do theirs.
Ceramic taps, especially the ones with a joystick, are much more trouble and expense.
If you want to get a plumber in, ask friends a neighbours for a local recommendation, A wrinkly old plumber with a cloth cap will be suitable. you do not need a gas engineer or an emergency call-out.
Do not use an advertising website masquerading as a recommendations website. Established local tradesmen have no need to advertise here as they are recommended by word of mouth. Do not ring a freephone or non-geographic number, it will be an agency.